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INTRODUCTION AND BACKGROUND The basic principle of evolution, survival of the fittest, was outlined by the naturalist Charles Darwin in his 1859 book On the origin of species. Though controversial at the time, the central ideas remain virtually unchanged to this date, even though much more is now known about the biological basis of inheritance. Darwin expressed his arguments verbally, but many attempts have since then been made to formalise the theory of evolution. The perhaps most well known are Population Genetics which aim to model the biological basis of inheritance but usually at the expense of ecological detail, Quantitative Genetics which incorporates quantitative traits influenced by genes at many loci and evolutionary game theory which ignores genetic detail but incorporates a high degree of ecological realism, in particular that the success of any given strategy depends on the frequency at which strategies are played in the population, a concept known as frequency dependence. Adaptive Dynamics is a set of techniques developed during the 1990s for understanding the long-term consequences of small mutations in the traits expressing the phenotype. They link Population Dynamics to Evolutionary Dynamics and incorporate and generalises the fundamental idea of frequency dependent selection from game theory. The number of papers using Adaptive Dynamics techniques is increasing steadily as Adaptive Dynamics is gaining ground as a versatile tool for evolutionary modelling. FUNDAMENTAL IDEAS Two fundamental ideas of Adaptive Dynamics are that the resident population can be assumed to be in a dynamical equilibrium when new mutants appear, and that the eventual fate of such mutants can be inferred from their initial growth rate when rare in the environment consisting of the resident. This rate is known as the invasion exponent when measured as the initial exponential growth rate of mutants, and as the basic Reproductive Number when it measures the expected total number of offspring that a mutant individual will produce in a lifetime. It can be thought of, and is indeed sometimes also referred to, as the invasion fitness of mutants. In order to make use of these ideas we require a mathematical model that explicitly incorporates the traits undergoing evolutionary change. The model should describe both the environment and the population dynamics given the environment, but in many cases the variable part of the environment consists only of the demography of the current population. We then determine the invasion exponent, the initial growth rate of a mutant invading the environment consisting of the resident. Depending on the model, this can be trivial or very difficult, but once determined, the Adaptive Dynamics techniques can be applied independent of the model structure. =Monomorphic evolution= A population consisting of individuals with the same trait is called monomorphic. If not explicitly stated differently, we will assume that the trait is a real number, and we will write r and m for the trait value of the monomorphic resident population and that of an invading mutant, respectively. INVASION EXPONENT AND SELECTION GRADIENT The invasion exponent is defined as the expected growth rate of an initially rare mutant in the environment set by the resident, which simply means the frequency of each phenotype (trait value) whenever this suffices to infer all other aspects of the equilibrium environment, such as the demographic composition and the availability of resources. For each r the invasion exponent can be thought of as the fitness landscape experienced by an initially rare mutant. The landscape changes with each successful invasion, as is the case in evolutionary game theory, but in contrast with the classical view of evolution as an optimisation process towards ever higher fitness. We will always assume that the resident is at its demographic attractor, and as a consequence for all r, as otherwise the population would grow indefinitely. The selection gradient is defined as the slope of the invasion exponent at , . If the sign of the invasion exponent is positive (negative) mutants with slightly higher (lower) trait values may successfully invade. This follows from the linear approximation :: which holds whenever . PAIRWISE-INVASIBILITY PLOTS The invasion exponent represents the fitness landscape as experienced by a rare mutant. In a large (infinite) population only mutants with trait values for which is positive are able to successfully invade. The generic outcome of an invasion is that the mutant replaces the resident, and the fitness landscape as experienced by a rare mutant changes. To determine the outcome of the resulting series of invasions pairwise-invasibility plots (PIPs) are often used. These show for each resident trait value all mutant trait values for which is positive. Note that is zero at the diagonal . In PIPs the fitness landscapes as experienced by a rare mutant correspond to the vertical lines where the resident trait value is constant. EVOLUTIONARILY SINGULAR STRATEGIES The selection gradient determines the direction of evolutionary change. If it is positive (negative) a mutant with a slightly higher (lower) trait-value will generically invade and replace the resident. But what will happen if vanishes? Seemingly evolution should come to a halt at such a point. While this is a possible outcome, the general situation is more complex. Traits
evolutionarily singular strategies. Near such points the fitness landscape as experienced by a rare mutant is locally `flat'. There are three qualitatively different ways in which this can occur. First, a degenerate case similar to the qubic where finite evolutionary steps would lead past the local 'flatness'. Second, a fitness maximum which is known as an evolutionarily stable strategy (ESS) and which, once established, cannot be invaded by nearby mutants. Third, a fitness minimum where disruptive selection will occur and the population branch into two morphs. This process is known as Evolutionary Branching . In a pairwise invasibility plot the singular strategies are found where the boundary of the region of positive invasion fitness intersects the diagonal. Singular strategies can be located and classified once the selection gradient is known. To locate singular strategies, it is sufficient to find the points for which the selection gradient
be classified then using the second derivative test from basic
(positive) the strategy represents a local fitness maximum (minimum).
If this does not hold the strategy is evolutionarily unstable and, provided that it also convergence stable, evolutionary branching will
stable monomorphic populations with slightly lower or slightly higher trait values must be invadable by mutants with trait values closer to
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